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IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inverses of banded matrices
Enrico Bozzojoint work with R. Bevilacqua, G. Del Corso, D. Fasino
Dipartimento di Informatica, Università di Pisa
INTERNATIONAL SEMINAR ON MATRIX METHODS ANDOPERATOR EQUATIONS
June 20 - 25, 2005, Moscow, Russia
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Banded matrices
Banded matrices are basic in matrix theory and inapplications and many authors studied the structure oftheir inverses.By assuming that the entries in the outermost diagonalsare nonzero it is possible to derive inversion formulasinvolving Schur complements.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Banded matrices
Banded matrices are basic in matrix theory and inapplications and many authors studied the structure oftheir inverses.By assuming that the entries in the outermost diagonalsare nonzero it is possible to derive inversion formulasinvolving Schur complements.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Schur complement
Let B be a n × n nonsingular banded matrix of the form
B =
(B11 LB21 B22
)where L is q × q lower triangular and nonsingular and p = n− qis the bandwidth. Let Ik be the k × k identity. Then
B =
(Iq O
B22L−1 Ip
) (O LS O
) (Ip O
L−1B11 Iq
),
where the p × p matrix S = B21 − B22L−1B11 is known as Schurcomplement of L in B and is nonsingular.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Schur complement
Let B be a n × n nonsingular banded matrix of the form
B =
(B11 LB21 B22
)where L is q × q lower triangular and nonsingular and p = n− qis the bandwidth. Let Ik be the k × k identity. Then
B =
(Iq O
B22L−1 Ip
) (O LS O
) (Ip O
L−1B11 Iq
),
where the p × p matrix S = B21 − B22L−1B11 is known as Schurcomplement of L in B and is nonsingular.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Schur complement
Let B be a n × n nonsingular banded matrix of the form
B =
(B11 LB21 B22
)where L is q × q lower triangular and nonsingular and p = n− qis the bandwidth. Let Ik be the k × k identity. Then
B =
(Iq O
B22L−1 Ip
) (O LS O
) (Ip O
L−1B11 Iq
),
where the p × p matrix S = B21 − B22L−1B11 is known as Schurcomplement of L in B and is nonsingular.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Example of inversion formula
This implies
B−1 =
(0 0
L−1 0
)+
(Ip
−L−1B11
)S−1 (
−B22L−1 Ip).
The submatrices of B−1 contained in the part of the matrix not“covered” by L−1 have rank not greater than the bandwidth p.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Example of inversion formula (continued)
If p = 1 then B is an unreduced lower Hessenberg and
B−1 =
(0 0
L−1 0
)+
1S
(1
−L−1B11
) (−B22L−1 1
).
The upper triangular part of B−1 is the upper triangular part of amatrix whose rank is one.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Asplund, 1959
Edgar Asplud in 1959 published the first result withoutassumptions besides nonsingularity.In particular, Asplund showed that if a nonsingular matrixcontains a null submatrix then its complementarysubmatrix in the inverse has a prescribed rank.The notion of complementary submatrices leads us todiscuss the nullity theorem that can be seen as ageneralization of Asplund’s result.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Asplund, 1959
Edgar Asplud in 1959 published the first result withoutassumptions besides nonsingularity.In particular, Asplund showed that if a nonsingular matrixcontains a null submatrix then its complementarysubmatrix in the inverse has a prescribed rank.The notion of complementary submatrices leads us todiscuss the nullity theorem that can be seen as ageneralization of Asplund’s result.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Asplund, 1959
Edgar Asplud in 1959 published the first result withoutassumptions besides nonsingularity.In particular, Asplund showed that if a nonsingular matrixcontains a null submatrix then its complementarysubmatrix in the inverse has a prescribed rank.The notion of complementary submatrices leads us todiscuss the nullity theorem that can be seen as ageneralization of Asplund’s result.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices
Let A be a square matrix of order n, let N = {1, 2, . . . , n} and letα and β be nontrivial subsets of N. Then we denote withA(α, β) the submatrix of A having row indices in α and columnindices in β.
DefinitionLet A and B be square matrices of order n. Then the twosubmatrices A(α, β) and B(N \ β, N \ α) are said to becomplementary.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices: example 1
In the matrix
A =
X X Y Y Y YX X Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrix A({1, 2}, {1, 2})is in red
In the matrix
B =
Y Y Y Y Y YY Y Y Y Y YY Y X X X XY Y X X X XY Y X X X XY Y X X X X
the submatrixB({3, 4, 5, 6}, {3, 4, 5, 6}) is inred
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices: example 1
In the matrix
A =
X X Y Y Y YX X Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrix A({1, 2}, {1, 2})is in red
In the matrix
B =
Y Y Y Y Y YY Y Y Y Y YY Y X X X XY Y X X X XY Y X X X XY Y X X X X
the submatrixB({3, 4, 5, 6}, {3, 4, 5, 6}) is inred
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices: example 2
In the matrix
A =
Y Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixA({1}, {3, 4, 5, 6}) is in red
In the matrix
B =
Y X X X X XY X X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixB({1, 2}, {2, 3, 4, 5, 6}) is in red
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices: example 2
In the matrix
A =
Y Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixA({1}, {3, 4, 5, 6}) is in red
In the matrix
B =
Y X X X X XY X X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixB({1, 2}, {2, 3, 4, 5, 6}) is in red
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices: example 3
In the matrix
A =
Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixA({1, 2}, {3, 4, 5, 6}) is in red
In the matrix
B =
Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixB({1, 2}, {3, 4, 5, 6}) is in red
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Complementary submatrices: example 3
In the matrix
A =
Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixA({1, 2}, {3, 4, 5, 6}) is in red
In the matrix
B =
Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y
the submatrixB({1, 2}, {3, 4, 5, 6}) is in red
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
The nullity theorem
Let null(·) denote the nullity, i.e., the dimension of the nullspace.
TheoremLet A be n × n and nonsingular. Then
null(A(α, β)) = null(A−1(N \ β, N \ α))
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Dr. Nullity and Mr. Rank
Let rk(·) denote the rank. For a matrix, not necessarily square,the sum of the rank and of the nullity equals the number ofcolumns.
CorollaryLet A be n × n and nonsingular. Then
rk(A−1(N \ β, N \ α)) = rk(A(α, β)) + n − (|α|+ |β|).
In the case where A(α, β) = O the corollary is exactly Asplundresult.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Application to banded matrices
If B is a nonsingular banded matrix with bandwidth p if wechoose
α = {1, . . . , k}β = {p + 1 + k , . . . , n} k = 1, . . . , n − p − 1
then B(α, β) = O and n − (|α|+ |β|) = p so that
rk(B−1(N \ β, N \ α)) = p.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
Definition
DefinitionLet A be a square matrix of order n, let N = {1, 2, . . . , n} and letΣ ⊆ N × N. The structured rank rk(A,Σ) is defined as themaximum of rk(A(α, β)) s.t. α× β ⊆ Σ.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Inverses of banded matricesThe nullity theoremStructured rank
An example of invariance under inversion
Let Σu = {(i , j)|i , j ∈ N ∧ j > i}. The “rank” version of the nullitytheorem allows to prove, for example, that
rk(A,Σu) = rk(A−1,Σu)
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
The Penrose Equations
1 AXA = A2 XAX = X3 (AX )∗ = AX4 (XA)∗ = XA
DefinitionGiven a matrix A let A{j , . . . , k} be the set of matrices whichsatisfy equations j , . . . , k among the four above. A matrixX ∈ A{j , . . . , k} is called an {j , . . . , k}-inverse of A.
For example A{1, 2, 3, 4} has exactly one element known asMoore-Penrose inverse of A.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
{1, 2}-inverses
If X ∈ A{1, 2} then:A ∈ X{1, 2};rk(X ) = rk(A);the nullspace of A and the range of X are supplementarysubspaces.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
{1, 2}-inverses
If X ∈ A{1, 2} then:A ∈ X{1, 2};rk(X ) = rk(A);the nullspace of A and the range of X are supplementarysubspaces.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
{1, 2}-inverses
If X ∈ A{1, 2} then:A ∈ X{1, 2};rk(X ) = rk(A);the nullspace of A and the range of X are supplementarysubspaces.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
A nullity theorem for {1, 2} inverses
TheoremLet A be n × n and X ∈ A{1, 2}. Then
|null(X (N \ β, N \ α))− null(A(α, β))| ≤ null(A) = null(X ).
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
The rank counterpart
Corollary
Let A be n × n and X ∈ A{1, 2}. Then
rk(X (N \ β, N \ α)) ≤ rk(A(α, β)) + n − (|α|+ |β|) + null(A)
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Application to banded matrices
If A is a banded matrix with bandwidth p and we choose
α = {1, . . . , k}β = {p + 1 + k , . . . , n} k = 1, . . . , n − p − 1
then A(α, β) = O and n − (|α|+ |β|) = p so that if X ∈ A{1, 2}then
rk(X (N \ β, N \ α)) ≤ p + null(A).
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Example
The matrix
Z =
0 0 0 01 0 0 00 1 0 00 0 1 0
is banded with bandwidth p = 2, 1, 0,−1.The matrices in Z{1, 2} have the form
a 1 0 0b 0 1 0c 0 0 1
da + eb + fc d e f
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Example (generalization)
Let 1 ≤ p ≤ n − 1 and let Zp be n × n defined as
Zp =
(O O
In−p O
)then the matrices in Zp{1, 2} have the form(
X In−pYX Y
)
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Generalized inversesThe new formulationApplications
Application to structured rank
The extended formulation of nullity theorem shows, forexample, that if A is square and B ∈ A{1, 2} then
|rk(A,Σu)− rk(B,Σu)| ≤ null(A)
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Outline
1 IntroductionInverses of banded matricesThe nullity theoremStructured rank
2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications
3 A formula for the Moore-Penrose inverse
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Assumptions
In order to simplify the presentation we assume that B is ann × n unreduced singular lower Hessenberg matrix. Recall that
B =
(In−1 O
B22L−1 1
) (O LS O
) (1 O
L−1B11 In−1
).
If B is singular S = 0 and setting P = L−1B11 and Q = B22L−1
we find
B =
(In−1Q
)L
(P In−1
).
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Formula for the Moore-Penrose inverse
B+ =
(O O
L−1 O
)+
P∗L−1Q∗
(1 + P∗P)(1 + QQ∗)
(1−P
) (−Q 1
)+
1(1 + P∗P)
(1−P
) (P∗L−1 O
)+
1(1 + QQ∗)
(O
L−1Q∗
) (−Q 1
).
The formula shows that the rank of the submatrices in theupper triangular part cannot exceed two.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Example
Let us consider the following tridiagonal matrix
T =
0 1 0 0 01 0 1 0 00 1 0 1 00 0 1 0 10 0 0 1 0
.
The matrix T is singular and null(T ) = 1.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Example (continued)
The Moore-Penrose inverse of T is
T + =13
0 2 0 −1 02 0 1 0 10 1 0 1 0
−1 0 −1 0 20 −1 0 2 0
.
The rank of every submatrix of T + above (or below) the maindiagonal is less or equal to 2.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Summary
We presented a new formulation of nullity theorem thatapplies to singular matrices.Generalized inverses of square banded matrices havestructured rank properties that recall those enjoyed byordinary inverses.We presented a formula for the Moore-Penrose inverse ofa singular banded matrix having nonzero outermostdiagonal.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Work in progress
Rectangular matricesA paper on this subject is under revision.What happens for infinite matrices?This would be an interesting research topic.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
Work in progress
Rectangular matricesA paper on this subject is under revision.What happens for infinite matrices?This would be an interesting research topic.
Bozzo et al. Generalized inverses of banded matrices
IntroductionA new formulation of the nullity theorem
A formula for the Moore-Penrose inverseSummary
References
E. AsplundInverses of matrices {aij} which satisfy aij = 0 for j > i + pMathematica Scandinavica, 7:57-60, 1959.
R. Bevilacqua, E. Bozzo, G. M. Del Corso, D. FasinoRank structure of generalized inverses of banded matricesReport, Dipartimento di Informatica, Università di Pisa,2005.
G. Strang, T. NguyenThe interplay of ranks of submatricesSIAM Review, 46:637-646, 2005.
Bozzo et al. Generalized inverses of banded matrices